Abstract: The statistical theory based on the parametric family of Rényi entropy functionals is a generalization of Gibbs statistics. Depending on the value of the involved parameter, the corresponding Rényi distribution can take both an exponential form and a power-law form, which is typical for a wide range of statistical models. In this paper, we prove the energy equipartition theorem in the case of Rényi statistics, which makes it possible to solve the problem of obtaining the average energy for a large number of classical statistical models. The proposed approach for calculating the average energy is compared with the procedure for directly calculating this quantity for a system described by the simplest power-low Hamiltonian. New relations are presented that simplify the calculations in the considered theory. A special case of the Rényi distribution, which represents a generalization of a power-low distribution and thus allows us to approximate some empirical data more precisely, has been studied.